Stochastic journeys of cell progenies through compartments and the role of self-renewal, symmetric and asymmetric division

Division and differentiation events by which cell populations with specific functions are generated often take place as part of a developmental programme, which can be represented by a sequence of compartments. A compartment is the set of cells with common characteristics; sharing, for instance, a spatial location or a phenotype. Differentiation events are transitions from one compartment to the next. Cells may also die or divide. We consider three different types of division events: (i) where both daughter cells inherit the mother’s phenotype (self-renewal), (ii) where only one of the daughters changes phenotype (asymmetric division), and (iii) where both daughters change phenotype (symmetric division). The self-renewal probability in each compartment determines whether the progeny of a single cell, moving through the sequence of compartments, is finite or grows without bound. We analyse the progeny stochastic dynamics with probability generating functions. In the case of self-renewal, by following one of the daughters after any division event, we may construct lifelines containing only one cell at any time. We analyse the number of divisions along such lines, and the compartment where lines terminate with a death event. Analysis and numerical simulations are applied to a five-compartment model of the gradual differentiation of hematopoietic stem cells and to a model of thymocyte development: from pre-double positive to single positive (SP) cells with a bifurcation to either SP4 or SP8 in the last compartment of the sequence.

where The first equation in the system (A.1) can be solved independently and the solution is in agreement with Eq. (5).Now, for i = 2, 3, . . ., N each equation of the system represents a linear first-order differential equation which requires solving the preceding equations.The general solution for each can be written as follows (see, for example, Ref. 1 ) ] e ∆ i t dt +Const i e −∆ i t , i = 2, . . ., N.
(A.3) Following this approach, the second equation takes the form which has the following general solution +Const 2 e −∆ 2 t .
For the initial condition and the solution for i = 2 reads .
Thus, in general for any i = 2, . . ., N, While we have considered here the case of different eigenvalues, the most general solution can be found in Ref 2 , which models the radioactive decay of a chain of nuclides.For illustrative purposes, the case where all eigenvalues are the same is studied next.

Derivation of Eq. (6)
If ∆ i = ∆ j = ∆ for all i, j ∈ {1, . . ., N}, system (A.1) for the irreversible model becomes Therefore, the solution of the second equation is given by Here, we make use of E[C C C 1 (t)] = e −∆t as in Eq. (A.2), and Const 2 = 0. Following a similar argument we can write in agreement with Eq. (6).The reader may notice that when substituting the exponential solution for the previous compartment, the product of exponentials in the integral on the right hand side is one.Thus, only the integration of a constant is required in each step; this generates the power t i−1 .

Derivation of Eq. (9)
Under the assumption ∆ i = ∆ j = ∆ for all i, j ∈ {1, . . ., N − 1}, but with ∆ N = 0, system (A.1) for the irreversible model becomes The differential equation for i = N is only a function of parameters and the variable relative to the previous compartment C N−1 ; thus, the solution reads We recall that for i = 1, . . ., N − 1 the solution is given by, Substituting for i = N − 1 and integrating, we obtain in agreement with Eq. ( 9).

B The progeny of a single progenitor cell: derivation of Eq. (16)
By the definition of the probability generating function, Ψ i (z) = E(z S i ).Moreover, we let E j ∈ {death, differentiation, self-renewal, asymmetric division, symmetric division}.We make use of Eq. (15), in particular +E(z S i | symmetric division) P(symmetric division).

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Now we observe that E(z S i | death) = z(Ψ i (z)) 0 as, in case of first event being death, the initial cell cannot move to the next compartment; E(z S i | differentiation) = z(Ψ i+1 (z)) as a single cell moves from compartment C i to the next compartment ) 2 as a cell in compartment C i divides and the two daughter cells stay in the same compartment as the mother; E(z S i | asymmetric division) = zΨ i (z)Ψ i+1 (z) as when one cell divides by asymmetric division, one of the two daughter cells moves to the next compartment and one stays in the same compartment as the mother; and E(z S i | symmetric division) = z(Ψ i+1 (z)) 2 as when a cell in compartment C i divides by symmetric division, the two daughter cells move to the following compartment C i+1 .Thus, we have Making use of events rates and their probabilities, we get Eq.( 16).

Derivation of τ i (lifespan)
We consider the average lifespan of a tracked cell starting in compartment C i ; by making use of first-step arguments we obtain the following linear system of equations, This can be written in a matrix form as follows where By application of the Thomas algorithm 3,4 , we can find the coefficients γi and ρi ; in particular, we obtain the following recursive system of linear equations This can be solved via backward substitution, which gives the general solution with γ1 and ρ1 defined in Section 2.3.1.

Derivation of η i : number of divisions on the lifeline
We now consider the number of division events along the lifespan of the tracked cell, starting in compartment C i .Using first-step arguments, we obtain the following system of linear equations for its average value

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This system can be written in matrix form as We can make use of the Thomas algorithm 3,4 to solve this tri-diagonal system of equations, and find the following recursive form Solving this system via backward substitution leads to the general solution with ρk and γp defined as in Section 2.3.2.

Derivation of ω
When looking at ω i (n), defined as the probability that a single cell starting in compartment C i divides exactly n times before it dies or leaves the system, for any non-negative integer n, we can define the associated linear system of equations.In particular, for illustrative purposes, let us consider n = 2, so that ω i (2) ≡ P(D i = 2) is the probability that a single cell starting in compartment C i divides exactly twice before it dies or leaves the system.In this case, the system written in matrix form reads as, This leads to the following recursive system , One can solve the system via backward substitutions which leads to the following solution for the case n = 2,

Derivation of β i ( j)
We now consider the probability of a tracked cell starting in compartment C i to die in a given compartment C j .In particular, and for illustrative purposes, let us consider β i (1), the probability of a tracked cell from compartment C i dying in compartment C 1 .Using a first-step argument, we can write the following linear system of equations, which can be written in matrix form as, . . .Analogous arguments can be applied to compute the probability of death of a chosen cell in any compartment C j , i.e., β i ( j), as shown in Section 2.3.3.